Computability-theoretic complexity of topological conjugacy in nV

Determine the exact computability-theoretic complexity of the topological conjugacy relation CP_3(nV) for each n≥2, beyond its established undecidability and analytic upper bound.

Background

The paper defines CP_3(G) as the relation consisting of pairs of elements of a group G acting on a Cantor space that are conjugate by an arbitrary homeomorphism of the underlying space. It proves that CP_3(nV) is undecidable for every n≥2, while CP_1(nV) and CP_2(nV) are shown to be Σ⁰₁-complete. The authors explain that a conjugating homeomorphism can be coded by compatible finite clopen partitions, yielding a Σ¹₁ upper bound for CP_3(nV), but their reduction does not establish a sharper classification.

The unresolved problem is therefore to identify the precise position of CP_3(nV) in the computability-theoretic hierarchy, including whether the analytic upper bound is optimal or can be improved.